> [!definition] > > Let $X$ be a [[Topological Space|topological space]] and $E \subset X$. A **partition of unity** on $E$ is a collection $\seqi{f} \subset C(X, [0, 1])$ of [[Space of Continuous Functions|continuous functions]] such that: > - For any $x \in X$, there exists $U \in \cn(x)$ such that $f_i \ne 0$ for finitely many $i$s. > - $\sum_{i \in I}f_i(x) = 1$ for all $x \in E$. > [!definition] > > Let $X$ be a topological space and $\seqi{f}$ be a partition of unity on $E$. $\seqi{f}$ is **subordinate** to an open cover $\mathcal U$ of $E$ if for each $i \in I$, there exists $U \in \mathcal U$ such that the [[Support of Function|support]] $\supp{f_i} \subset U$.